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<H2><A NAME="SECTION03242000000000000000"></A>
<A NAME="subseccomporthog"></A>
<BR>
Orthogonal Factorizations and Linear Least Squares Problems
</H2>

<P>
LAPACK provides a number of routines for factorizing a general
rectangular <I>m</I>-by-<I>n</I> matrix <I>A</I>,
as the product of an <B>orthogonal</B> matrix (<B>unitary</B> if complex)
and a <B>triangular</B> (or possibly trapezoidal) matrix.

<P>
A real matrix <I>Q</I> is <B>orthogonal</B> if <I>Q</I><SUP><I>T</I></SUP> <I>Q</I> = <I>I</I>;
a complex matrix <I>Q</I> is <B>unitary</B> if <I>Q</I><SUP><I>H</I></SUP> <I>Q</I> = <I>I</I>.
Orthogonal or unitary matrices have the important property that they leave the
two-norm of a vector invariant:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\|x||_2 = \|Qx\|_2, \quad \mbox{if $Q$\  is orthogonal or unitary.}
\end{displaymath}
 -->


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 WIDTH="351" HEIGHT="31" BORDER="0"
 SRC="img101.gif"
 ALT="\begin{displaymath}
\Vert x\vert\vert _2 = \Vert Qx\Vert _2, \quad \mbox{if $Q$\ is orthogonal or unitary.}
\end{displaymath}">
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<P></P>
As a result, they help to maintain numerical stability because they do not
<A NAME="2563"></A>
amplify rounding errors.

<P>
Orthogonal factorizations<A NAME="2564"></A> are used in
the solution of linear least squares problems<A NAME="2565"></A>.
They may also be used to perform preliminary
steps in the solution of eigenvalue or
singular value problems.

<P>
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<!--Table of Child-Links-->
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<UL>
<LI><A NAME="tex2html4693"
 HREF="node40.html"><I>QR</I> Factorization</A>
<LI><A NAME="tex2html4694"
 HREF="node41.html"><B><I>LQ</I></B> Factorization</A>
<LI><A NAME="tex2html4695"
 HREF="node42.html"><B><I>QR</I></B> Factorization with Column Pivoting</A>
<LI><A NAME="tex2html4696"
 HREF="node43.html">Complete Orthogonal Factorization</A>
<LI><A NAME="tex2html4697"
 HREF="node44.html">Other Factorizations</A>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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